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Expanding triple (cross) product

  1. Oct 6, 2013 #1
    1. The problem statement, all variables and given/known data

    Use (the bac-cab rule) to expand this triple product:L = mr x (ω x r)

    If r is perpendicular to ω, show that you obtain the elementary formula, angular momentum = mvr.

    (The bold letters are vectors.)

    2. Relevant equations

    A X (B X C) = (A[itex]\cdot[/itex]C)B - (A[itex]\cdot[/itex]B)C

    3. The attempt at a solution

    Well, simply doing the bac-cab rule, I get

    (mr [itex]\cdot[/itex]ω)r - (mr [itex]\cdot[/itex]r)ω

    Which isn't even close to the answer in the book, which is:

    L=m[r2ω-(ω [itex]\cdot[/itex]r)r]

    No idea how they got that.

    Even if I distribute the r, they don't have it distributed to the ω, and on the right term, they don't distribute the ω. I don't understand.
    Thanks
     
  2. jcsd
  3. Oct 6, 2013 #2
    This is wrong, going by the rule we get
    m[(r.r)ω-(r.ω)r]
    On simplification we'll get the answer.....
    Regards
     
  4. Oct 6, 2013 #3
    OK yeah I lost track of which one was A, B, and C.

    But this is what I get:
    (mr[itex]\cdot[/itex]r)ω-(mr[itex]\cdot[/itex]ω)r

    r[itex]\cdot[/itex]r simplifies to r2 because it's the vector dotted into itself, which produces the magnitude of that vector squared?

    And according to the solution, why didn't they distribute the r through the parentheses in the second term?

    Thanks.
     
  5. Oct 6, 2013 #4

    vela

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    What specifically do you mean by "distribute the r through the parentheses"?
     
  6. Oct 6, 2013 #5
    Oh ok, so that r isn't allowed to go into the parentheses until the r is dotted into the ω?
     
  7. Oct 6, 2013 #6

    vela

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    I guess I still don't know what you're trying to do. ##\vec{\omega}\cdot\vec{r}## is a scalar, and the result of the product multiplies ##\vec{r}##. By pulling ##\vec{r}## into the parentheses, what do you intend to accomplish?
     
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